Linear bosonic quantum channels defined by superpositions of maximally distinguishable Gaussian environments

A minimal energy quantum superposition of two maximally distinguishable, isoenergetic single mode Gaussian states is used to construct the system-environment representation of a class of linear bosonic quantum channels acting on a single bosonic mode. The quantum channels are further defined by unitary dynamics of the system and environment corresponding to either a passive linear optical element $U_{\text{BS}}$ or two-mode squeezing $U_{\text{TM}}$. The notion of nonclassicality distance is used to show that the initial environment superposition state becomes maximally nonclassical as the constraint energy is increased. When the system is initially prepared in a coherent state, application of the quantum channel defined by $U_{\text{BS}}$ results in a nonclassical state for all values of the environment energy constraint. We also discuss the following properties of the quantum channels:

1. the maximal noise that a coherent system can tolerate, beyond which the linear bosonic attenuator channel defined by $U_{\text{BS}}$ cannot impart nonclassical correlations to the system, 2) the noise added to a coherent system by the phase-preserving linear amplification channel defined by $U_{\text{TM}}$, and 3) a generic lower bound for the trace norm contraction coefficient on the closed, convex hull of energy-constrained Gaussian states.